Linear separability

The amount of branching introduced into a polymer is an additional variable that must be specified for the molecule to be fully characterized. When only a slight degree of branching is present, the concentration of junction points is sufficiently low that these may be simply related to the number of chain ends. For example, two separate linear molecules have a total of four ends. If the end of one of these linear molecules attaches itself to the middle of the other to form a T, the resulting molecule has three ends. It is easy to generalize this result. If a molecule has v branches, it has v 2 chain ends if the branching is relatively low. Branched molecules are sometimes described as either combs or... 

ENSORB [ExxoN adSORBtion] A process for separating linear from branched hydrocarbons, using a zeolite molecular sieve. The adsorbed gases are desorbed using ammonia. It operates in a cyclic, not a continuous, mode. Developed by Exxon Research Engineering Company, and used by that company on a large scale at the Exxon refinery in Baytown, TX. Asher, W. J., Campbell, M. L., Epperly, W. R., and Robertson, J. LHydrocarbon Process., 1969, 48(1), 134. 

IsoSiv [Isomer separation by molecular sieves] A process for separating linear hydrocarbons from naphtha and kerosene petroleum fractions. It operates in the vapor phase and uses a modified 5A zeolite molecular sieve, which selectively adsorbs linear hydrocarbons, excluding branched ones. Developed by Union Carbide Corporation and widely licensed, now by UOP. The first plant was operated in Texas in 1961. By 1990, more than 30 units had been licensed worldwide. See also Total Isomerization. 

Molex A version of the Sorbex process, for separating linear aliphatic hydrocarbons from branched-chain and cyclic hydrocarbons in naphtha, kerosene, or gas oil. The process operates in the liquid phase and the adsorbent is a modified 5A zeolite the pores in this zeolite will admit only the linear hydrocarbons, so the separation factor is very large. First commercialized in 1964 by 1992, 33 plants had been licensed worldwide. See also Parex (2). 

N-ISELF A process for separating linear hydrocarbons from light naphtha by selective adsorption on a zeolite. Developed by Societe Nationale Elf-Aquitaine, France. 

Parex (2) [Paraffin extraction] A process for separating linear aliphatic hydrocarbons from branched-chain and cyclic hydrocarbons by means of a zeolite 5 A adsorbent. The products are desorbed with a mixture of steam and ammonia. Developed in the mid-1960s by Luena-Werke and Schwedt in East Germany and operated in East Germany, Bulgaria and the USSR. Broadly similar to Molex and not to be confused with Paiex (1). 

TSF [Texaco Selective Finishing] A process for separating linear from branched-chain aliphatic hydrocarbons by PSA, using zeolite 5A as the adsorbent. The desorbent is a hydrocarbon having two to four carbon atoms less than the feed. Developed by Texaco in the late 1950s. Believed to be still in operation in its Trinidad refinery as of 1990. 

B is the magnetic field strength and, as always, A denotes an isotopic difference. Equation 8.9 shows that isotopes of different masses will move in paths with radii R /R = (M /M)1/2. For a 180° sector like that in Fig. 8.6 the two isotopes are separated linearly by... 

It is also important to appreciate that adequate reproducibility is not the same as absolute reproducibility unless your assays employ isocratic elution. Low levels of variation among different lots of media can be offset by the use of linear gradient separations. Linear gradients are good insurance with HIC assays in any case because they also buffer routine variations in mobile-phase composition or ambient temperature. 

This system amounts to two separate linear systems of equations. The first block row in... 

It has to be remarked that the model discussed above is valid for the description of linear relationships between concentration and the response. If the concentration range has to be extended further, nonlinear effects cannot be excluded [ZWANZIGER et al., 1988]. Then the whole concentration range should be split into two experimental plans for separate treatment of the high and the low concentration levels. Accordingly, two separate linear models can be calculated for the different experimental plans. 

The data for the expanded and contracted minerals plot as two separate linear relations with contracted clays having larger tetrahedral rotation values for given oct.Atet. values than the expanded clays. This is presumably due to the K which aids the tetrahedral rotation in the contracted clays. 

Liquid chromatography was used by Rho and Choi [46] for the simultaneous determination of calcium and magnesium. A column of 4.6mm by 25cm, containing Zipax SCX was used for the separation. Linear calibration curves were obtained in the range of 1x 10-4 to 5x 10 4M and the correlation coefficient of the calibration curves was in the range of 0.9952-0.9996. 

To calculate characteristics of linear viscoelasticity, one can consider linear approximation of constitutive relations derived in the previous section. The expression (9.19) for stress tensor has linear form in internal variables x"k and u"k, so that one has to separate linear terms in relaxation equations for the internal variables. This has to be considered separately for weakly and strongly entangled system. 

In two later sections, we will deal with numerical integration, which is required to solve the differential equations for complex mechanisms. Before that, we will describe nonlinear fitting algorithms that are significantly more powerful and faster than the direct-search simplex algorithm used by the MATLAB function fminsearch. Of course, the principle of separating linear (A) and nonlinear parameters (k) will still be applied. 

It is important to recall at this point that k comprises only the nonlinear parameters, i.e., the rate constants. The linear parameters, i.e., the elements of the matrix A containing the molar absorptivities, are solved in a separate linear regression step, as described earlier in Equation 7.9 and Equation 7.10. 

The lattice parameters of nonstoichiometric uranium oxides, quenched from 1100° C., were determined within the composition range U02 to U4Og. Two separate linear relations for the lattice parameter as a function of oxygen content were obtained one characteristic of U02+2. and the other of U409 J/. The two functions are a0 = 5.4705 - 0.094 x (0 < x < 0.125) and a0 = 5.4423 + 0.029 y (0 < y < 0.31). Helium displacement densities were determined for some samples the values obtained are consistent with an oxygen interstitial model for U02+a. and an oxygen vacancy model for U409 y ... 

Other examples of successful combinations of liquid chromatography and MALDI-TOF have been reported by Kruger et al. who separated linear and cyclic fractions of polylactides by LC-CC [184]. Just et al. were able to separate cyclic siloxanes from linear silanols and to characterize their chemical composition [185]. The calibration of an SEC system by MALDI-TOF was discussed by Mon-taudo et al. [186]. Poly( dimethyl siloxane) (PDMS) was fractionated by SEC into different molar mass fractions. These fractions were subjected to MALDI-TOF for molar mass determination. The resulting peak maximum molar masses were... 

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